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This article has two main points. The principal theme is that even the most abstract ideas, those of pure mathematics, arise from experience of the real world. Yet the dominant views of mathematics will be shown to be varieties of idealism: they all regard concepts like numbers and equations as having an existence or meaning quite independent of the real world. Each of these views will be shown to lead to considerable problems. Only materialism can give a coherent picture of mathematics, as a body of ideas that has arisen from the world and which has developed to solve real problems facing society. The second theme is to look at formal reasoning, like mathematics and formal logic, to assess the strengths and limitations of this method and to compare it with the dialectic framework. It will be argued that formal reasoning and dialectics are not rival alternatives but complementary. Both are necessary components of a fully scientific method of understanding the world.

First let us briefly sketch the main views of mathematics.